Both Ice House Resort and Robb's Resort allow leaving/parking your tow trailer or rig. Ice House charges about $5/night to keep an eye on your rig. Ice House is more accessible to folks coming from the Placerville area. This is your only gas stop near the trail (on the mountain, as we say). More here.
Robb's Resort is online (click here). They have a restaurant and camping facilities, but are no longer able to sell gas (read more here). I'm not sure what they charge to leave your tow rig there. They are closer to the trail head than Ice House.
On the Tahoe side, your best bet is the Tahoma (Homewood) Staging Area right at the entrance to the trail. Barker Pass access and OHV area is another option for storage on this end.